One of the most famous pieces of music ever written for the trumpet is Haydn's Trumpet Concerto. It is so famous because it was the first piece of music ever written for chromatic trumpet (the type of trumpets we have today).
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A Solo Concerto is a concerto i which a single soloist is accompanied by an orchestra. It is the most frequent type of concerto. It originated in the Baroque Period (approx. 1600 - 1750) as an alternative to the traditional Concertino in a concerto grosso. Concerto = an instrumental work that maintains contrast between an orchestral ensemble and a smaller group or a solo instrument, or among various groups of an undivided orchestra. (fast-slow-fast) three movements. Concertino = The group of solo instruments in a concerto grosso. Concerto Grosso = a musical form, common in the Baroque period, in which contrasting sections are played by full orchestra and by a small group of soloists. Hope that helps :)
That describes a concerto.
Typically a concerto grosso is for a small ensemble of solo instruments, rather than a soloist, with a regular orchestra. While concerti grossi were composed almost exclusively during the baroque period, a few composers in later periods, including our own, have written in the same form.
He did not write the following types of concertos: viola, double bass, oboe, flute, clarinet, harp, and many more instruments. The only concertos that he has written are: Bassoon Concerto, Piano Concerto, Violin Concerto, Cello Concerto.